Problem: A section is cut out of a circular piece of paper having radius four inches, as shown. Points A and B are then glued together to form a right circular cone. What is the circumference of the base of the resulting cone? Express your answer in terms of $\pi$.  (The $270^\circ$ sector forms the cone.)

[asy]import graph;
draw(Circle((0,0),42.4),linewidth(1));
draw((0,0)--(30,30),linewidth(2));
draw((0,0)--(30,-30),linewidth(2));
label("$A$",(30,30),E);
label("$B$",(30,-30),E);
label("4''",(15,15),NW);
draw((3,3)--(6,0));
draw((6,0)--(3,-3));

[/asy]
Explanation: The circumference of the whole circle is $2 \pi \cdot 4 = 8 \pi$.  Then the circumference of the base of the cone is \[\frac{270^\circ}{360^\circ} \cdot 8 \pi = \boxed{6 \pi}.\]